Excitons are quasi-particles that form when Coulombinteracting electrons and holes in semiconductors are bound into pair states. They have many features analogous to those of atomic hydrogen. Because of this, researchers are interested in exploring excitonic phenomena, from optical, quantum-optical and thermodynamic transitions to the possible condensation of excitons into a quantum-degenerate state. Excitonic signatures commonly appear in the optical absorption and emission of direct-gap semiconductor systems. However, the precise properties of incoherent exciton populations in such systems are difficult to determine and are the subject of intense debate. We review recent contributions to this discussion, and argue that to obtain detailed information about exciton populations, conventional experimental techniques should be supplemented by direct quasi-particle spectroscopy using the relatively newly …